Evaluate the integrals in Exercises 33–54.
∫₀^(π/4) (1 + e^(tan θ)) sec²θ dθ

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Identify the integral to be evaluated: \(\int_0^{\frac{\pi}{4}} (1 + e^{\tan \theta}) \sec^2 \theta \, d\theta\).

Recognize that the integrand contains \(\sec^2 \theta\) and \(e^{\tan \theta}\), suggesting a substitution involving \(\tan \theta\) because the derivative of \(\tan \theta\) is \(\sec^2 \theta\).

Let \(u = \tan \theta\). Then, compute \(du = \sec^2 \theta \, d\theta\), which means \(\sec^2 \theta \, d\theta = du\).

Change the limits of integration from \(\theta\) to \(u\): when \(\theta = 0\), \(u = \tan 0 = 0\); when \(\theta = \frac{\pi}{4}\), \(u = \tan \frac{\pi}{4} = 1\).

Rewrite the integral in terms of \(u\): \(\int_0^1 (1 + e^u) \, du\). Then, split the integral into two simpler integrals: \(\int_0^1 1 \, du + \int_0^1 e^u \, du\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a function and its derivative.

Derivative of the Tangent Function

The derivative of tan(θ) with respect to θ is sec²(θ). Recognizing this relationship helps in substitution because if the integrand contains sec²(θ) dθ, it can be replaced by d(tan θ), simplifying the integral.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two limits. When performing substitution, the limits of integration must be adjusted to correspond to the new variable, ensuring the integral is evaluated correctly within the new bounds.

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